To the Problem of Magnetic Fields and Spin Polarization of

Nucleons in Astrophysics

A.I. Sery,

Brest State A.S. Pushkin University (Brest, Belarus)

XII Gomel School «Actual Problems of Microworld Physics» 01.08.2013

I. The object and subject of investigation

p- and np-gas in astrophysics

np-mixture at SN-II (with short lifetime)(Bruenn, S.W. Recent 2D/3D Core-Collapse Supernovae Simulations Results Obtained with the CHIMERA Code /

S.W. Bruenn et al. //physics.fau.edu)

Possible states of H in WD and SN-II

In spite of that, the models of degenerate ep- and enp-gases should be also considered as an approximation

Spontaneous nucleon spin polarization (leading to magnetization) can be based on nuclear

pseudomagnetism, exchange and correlation corrections. What is important here:

• V.G. Baryshevsky and M.I. Podgoretzky predicted nuclear pseudomagnetism in 1964; After that V. G. Baryshevsky proposed the idea that spin-polarized state of nucleon system can turn out energetically preferrable in comparison with non-polarized one; Experiments in 1970s showed that nuclear pseudomagnetic field exceeds ordinary “polarizational” magnetic field by 2 orders;

• Is spontaneous spin polarization possible at SN II explosions?

• Terrestrial ferromagnetism is mainly due to exchange energy of electrons; at astrophysical densities it seems to be impossible;

• Is ferromagnetism possible for proton gas due to exchange energy of protons in astrophysics?

Basic known explanations of MF of WD and SN-II.

New questions for WD (DAP, DAH) and SN-II:

• Is ferromagnetism of nucleons possible?• Can it be an additional explanation of magnetic

fields?

Some papers on MF of WD• 1. Angel, J.P.R. Magnetic White Dwarfs // J.P.R. Angel. // Ann. Rev. Astron.

Astrophys. – 1978. – Vol. 16 – P. 487.• 2. Landstreet, J.D. The Magnetic Fields of Single White Dwarfs / J.D. Landstreet //

White Dwarfs and Variable Degenerate Stars (IAU Colloq. No 53). Eds. H.M. Van Horn and V. Weidemann. – Rochester, New York: University of Rochester Press, 1979.

• 3. Valyavin, G.G. Magnetic fields of white dwarfs // G.G. Valyavin, T.E. Burlakova, S.N. Fabrika and D.N. Monin // Astronomy Reports. – 2003. – Vol. 47, № 7 – P. 587–599.

• 4. Jordan, S. The fraction of DA white dwarfs with kilo-Gauss magnetic fields. / S. Jordan, R. Aznar Cuadrado, R. Napiwotzki, H.M. Schmid, S.K. Solanki // arxiv.org/abs/astro-ph/0610875v2 – P. 1–10.

• 5. Cumming, Andrew. Magnetic Field Evolution in Accreting White Dwarfs / Andrew Cumming // ASP Conference Series. – P. 1–13.

• 6. Tout, Christopher A. The origin of the strongest magnetic fields in dwarfs. / Christopher A. Tout // Pramana – J. Phys. – July 2011. – Vol. 77, № 1. – P. 199–212.

• 7. Liebert, James. The True Incidence of Magnetism Among Field White Dwarfs. / James Liebert, P. Bergeron and J.B. Holberg // arxiv.org/abs/astro-ph/0210319v2 – P. 1–16.

• 8. Nordhaus, Jason. Formation of high-field magnetic white dwarfs from common envelopes. / Jason Nordhaus [et al.] // www.pnas.org/cgi/doi/10.1073/pnas.1015005108 – P. 1–6.

• 9. Norton, A.J. The spin periods and magnetic moments of white dwarfs in magnetic cataclysmic variables / A.J. Norton, G.A. Wynn, and R.V. Somerscales // Astrophysical Journal. – 2004. – 614(1, Par). – P. 349–357.

Some papers on MF of SN

• 1. Steven A. Balbus and John F. Hawley. Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70, 1–53 (1998)

• 2. Shizuka Akiyama and J. Craig Wheeler. The Nonmonotonic Dependence of Supernova and Compact Remnant Formation on Progenitor Rotation. The Astrophysical Journal, 629:414-421, 2005 August 10

II. Methods and approximations

Restrictions imposed on our model

Metallization density (perhaps, overestimated)

Dirac delta, or Fermi pseudopotential, is used for nuclear interaction (it is enough for

densities considered here)

Calculations for p- or np-Stoner criterion (1 non-degenerate or mixture of 2 degenerate

Fermi-gases): special algorithm was developed on the basis of the algorithm for 1

degenerate Fermi-gas:

• L.S. Levitov, A.V. Shitov. Green Functions. Tasks and solutions. Moscow, Fizmatlit, 2003 [in Russian].

Stoner (ferromagnetism) criterion formulations

III. Stoner criterion for degenerate np-mixture

Energy density terms for degenerate np-gas

Using the methods mentioned above, we obtain Stoner criterion (w<0) for degenerate np-gas

What is possible in WDs (left) and at SN-II explosions (left and right); ferromagnetism is above the curves

Corresponding FM densities (table)

Magnetic fields of SN (though it’s better to consider non-degenerate np-system

The main conclusions on Part III• Spontaneous spin polarization for degenerate np-Fermi-

gas is, most likely, possible• It can occur right after a SN II explosion, though np-

mixture in SN II is, most likely, non-degenerate• It could be also possible in WD under neutronization

threshold if the medium in WD was not crystal• There is only 1 region or ferromagnetism on (np,nn)-

diagram if only nuclear interaction is considered• There are 2 regions or ferromagnetism on (np,nn)-

diagram if not only nuclear, but also Coulomb exchange interaction are considered

• Coulomb correlation energy reduces the positive effect of Coulomb exchange energy, but net effect for ferromagnetism is still more favourable in comparison with the case when only nuclear energy is considered

You can find the details of the calculations for part III in the article

(http://www.brsu.by/science/vestnik-brgu)

IV. Stoner criterion for non-degenerate hydrogen

The condition of non-degeneration

Chemical potential of ideal gas

Energy density of hot ideal Fermi-gas

Exchange (Coulomb) energy density

Correlation (Coulomb) energy density (according to Vedenov and

Larkin formula)

Nuclear energy density

After some rearrangements we obtain Stoner

criterion (w<0) for non-degenerate hydrogen

Magnetic fields of WD

As for SN-II, it’s better to consider non-degenerate np-system because here B ~ 108 Gs

is predicted for pure H in SN-II

The main conclusions on Part IV• Spontaneous spin polarization for hot proton gas in

outer layers of WDs and SN II is possible• The density region of ferromagnetism is smaller in

comparison with degenerate hydrogen• Magnetic fields of B~106-109 Gs probably cannot

be explained directly by proton spin polarization though such a polarization can form initial magnetic field B~104 Gs which can be amplified by other mechanisms

• The question of magnetization at SN-II explosions (in outer layers) should be considered for non-degenerate np-system because only B~108 Gs is predicted for pure hydrogen

You can find the details of the calculations for part IV in the article

(http://www.brsu.by/science/vestnik-brgu)

Thank you for your attention !